In order to compute the average energy, therefore, one needs to be able to compute the average of the potential \(\langle U \rangle \). In general, this is a nontrivial task, however, let us work out the average for the case of a pairwise-additive potential of the form

i.e., U is a sum of terms that depend only the distance between two particles at a time. This form turns out to be an excellent approximation in many cases. U therefore contains N(N-1) total terms, and \(\langle U \rangle \) becomes

Thus, we have an expression for E in terms of a simple integral over the pair potential form and the radial distribution function. It also makes explicit the deviation from ``ideal gas'' behavior, where E=3NkT/2.

By a similar procedure, we can develop an equation for the pressure P in terms of g(r). Recall that the pressure is given by

\[ P = {1 \over \beta } {\partial \ln Q \over \partial V} \]

\[ = {1 \over \beta Z_N}{\partial Z_N \over \partial \beta } \]

The volume dependence can be made explicit by changing variables of integration in \(Z_N \) to

which again gives a simple expression for the pressure in terms only of the derivative of the pair potential form and the radial distribution function. It also shows explicitly how the equation of state differs from the that of the ideal gas \({P \over kT } = \rho \).

From the definition of g(r) it can be seen that it depends on the density \(\rho \) and temperature T: \(g (r) = g (r; \rho , T ) \). Note, however, that the equation of state, derived above, has the general form

\[ {P \over kT } = \rho + B \rho ^2 \]

which looks like the first few terms in an expansion about ideal gas behavior. This suggests that it may be possible to develop a general expansion in all powers of the density \(\rho \) about ideal gas behavior. Consider representing \(g (r; \rho, T ) \) as such a power series:

are known as the virial coefficients. The coefficient \(B_2 (T) \) is of particular interest, as it gives the leading order deviation from ideal gas behavior. It is known as the second virial coefficient. In the low density limit, \(g (r; \rho , T) \approx g_0 (r; T) \) and \(B_2 (T) \) is directly related to the radial distribution function.